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A three-step defect-correction algorithm for incompressible flows with friction boundary conditions

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Abstract

Based on finite element discretization and a recent variational multiscale-stabilized method, we propose a three-step defect-correction algorithm for solving the stationary incompressible Navier-Stokes equations with large Reynolds numbers, where nonlinear slip boundary conditions of friction type are considered. This proposed algorithm consists of solving one nonlinear Navier-Stokes type variational inequality problem on a coarse grid in a defect step, and solving two stabilized and linearized Navier-Stokes type variational inequality problems which have the same stiffness matrices with only different right-hand sides on a fine grid in correction steps. In the defect step, an artificial viscosity term is used as a stabilizing factor, making the nonlinear system easier to solve. Error bounds of the approximate solutions in L2 norms for the velocity gradient and pressure are estimated. Scalings of the algorithmic parameters are derived. Some numerical results are given to support the theoretical predictions and test the validity of the present algorithm.

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References

  1. Fujita, H.: Flow Problems with Unilateral Boundary Conditions. Lecons, Collège de France (1993)

    Google Scholar 

  2. Fujita, H., Kawarada, H.: Variational inequalities for the Stokes equation with boundary conditions of friction type. Recent Dev. Domain Decompos. Methods Flow Probl. 11, 15–33 (1998)

  3. Fujita, H.: A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. RIMS Kokyuroku 888, 199–216 (1994)

    MATH  MathSciNet  Google Scholar 

  4. Fujita, H.: Non-stationary Stokes flows under leak boundary conditions of friction type. J. Comput. Math. 19, 1–8 (2001)

    MATH  MathSciNet  Google Scholar 

  5. Fujita, H.: A coherent analysis of Stokes flows under boundary conditions of friction type. J. Comput. Appl. Math. 149, 57–69 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Li, Y., Li, K. T.: Existence of the solution to stationary Navier-Stokes equations with nonlinear slip boundary conditions. J. Math. Anal. Appl. 381, 1–9 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Li, Y., Li, K. T.: Global strong solutions of two-dimensional Navier-Stokes equations with nonlinear slip boundary conditions. J. Math. Anal. Appl. 393, 1–13 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kashiwabara, T.: On a strong solution of the non-stationary Navier-Stokes equations under slip or leak boundary conditions of friction type. J. Differ. Equ. 254, 756–778 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kashiwabara, T.: Finite element method for Stokes equations under leak boundary condition of friction type. SIAM J. Numer. Anal. 51, 2448–2469 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Li, Y., An, R.: Semi-discrete stabilized finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions based on regularization procedure. Numer. Math. 117, 1–36 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Li, Y., An, R.: Penalty finite element method for Navier-Stokes equations with nonlinear slip boundary conditions. Int. J. Numer. Methods Fluids 69, 550–566 (2012)

    Article  MathSciNet  Google Scholar 

  12. An, R.: Comparisons of Stokes/Oseen/Newton iteration methods for Navier-Stokes equations with friction boundary conditions. Appl. Math. Model. 38, 5535–5544 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  13. Li, Y., An, R.: Two-level variational multiscale finite element methods for Navier-Stokes type variational inequality problem. J. Comput. Appl. Math. 290, 656–669 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  14. Qiu, H. L., Mei, L. Q., Zhang, Y. C.: Two-grid variational multiscale algorithms for the stationary incompressible Navier-Stokes equations with friction boundary conditions. Numer. Methods Partial Differ. Equ. 33, 546–569 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jing, F. F., Li, J., Chen, Z. X., Zhang, Z. H.: Numerical analysis of a characteristic stabilized finite element method for the time-dependent Navier-Stokes equations with nonlinear slip boundary conditions. J. Comput. Appl. Math. 320, 43–60 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  16. Zhou, K.R., Shang, Y.Q.: Parallel iterative stabilized finite element algorithms for the Navier-Stokes equations with nonlinear slip boundary conditions. Int. J. Numer. Methods Fluids 93, 1074–1109 (2021)

  17. John, V.: Slip with friction and penetration with resistance boundary conditions for the Navier-Stokes equations–numerical tests and aspects of the implementation. J. Comput. Appl. Math. 147, 287–300 (2002)

  18. Axelsson, O., Layton, W.: Defect correction methods for convection dominated, convection diffusion equations. RAIRO J. Numer. Anal. 24, 423–455 (1990)

    MATH  Google Scholar 

  19. Ervin, V. J., Layton, W., Maubach, J. M.: Adaptive defect-correction methods for viscous incompressible flow problems. SIAM J. Numer. Anal. 37, 1165–1185 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  20. Layton, W., Lee, H., Peterson, J.: A defect-correction method for the incompressible Navier-Stokes equations. Appl. Math. Comput. 129, 1–19 (2002)

    MATH  MathSciNet  Google Scholar 

  21. Wang, K.: A new defect correction method for the Navier-Stokes equations at high Reynolds numbers. Appl. Math. Comput. 216, 3252–3264 (2010)

    MATH  MathSciNet  Google Scholar 

  22. Si, Z. Y.: Second order modified method of characteristics mixed defect-correction finite element method for time dependent Navier-Stokes problems. Numer. Algor. 59, 271–300 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  23. Shang, Y. Q.: Parallel defect-correction algorithms based on finite element discretization for the Navier-Stokes equations. Comput. Fluids 79, 200–212 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  24. Huang, P.Z., Feng, X.L., He, Y.N.: Two-level defect-correction Oseen iterative stabilized finite element methods for the stationary Navier-Stokes equations. Appl. Math. Model. 37, 728–741 (2013)

  25. Qiu, H.L., Mei, L.Q., Liu, H., Cartwright, S.: A defect-correction stabilized finite element method for Navier-Stokes equations with friction boundary conditions. Appl. Numer. Math. 90, 9–21 (2015)

  26. Qiu, H. L., Mei, L. Q.: Two-level defect-correction stabilized finite element method for Navier-Stokes equations with friction boundary conditions. J. Comput. Appl. Math. 280, 80–93 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  27. Liu, A., Li, Y., An, R.: Two-level defect-correction method for steady Navier-Stokes problem with friction boundary conditions. Adv. Appl. Math. Mech. 8, 932–952 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  28. Xu, J. C.: A novel two-grid method for semilinaer elliptic equations. SIAM J. Sci. Comput. 15, 231–237 (1994)

    Article  MathSciNet  Google Scholar 

  29. Xu, J. C.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33, 1759–1777 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  30. Layton, W.: A two level discretization method for the Navier-Stokes equations. Comput. Math. Appl. 5, 33–38 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  31. Layton, W., Lee, H. K., Peterson, J.: Numerical solution of the stationary Navier-Stokes equations using a multilevel finite element method. SIAM J. Sci. Comput. 20, 1–12 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  32. Layton, W., Lenferink, H. W. J.: A multilevel mesh independence principle for the Navier-Stokes equations. SIAM J. Numer. Anal. 33, 17–30 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  33. Girault, V., Lions, J. L.: Two-grid finite element scheme for the transient Navier-Stokes problem. Math. Model. Numer. Anal. 35, 945–980 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  34. Franca, L. P., Nesliturk, A.: On a two-level finite elemnt method for the incompressible Navier-Stokes equations. Int. J. Numer. Methods Engrg. 52, 433–453 (2001)

    Article  MATH  Google Scholar 

  35. He, Y. N.: Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal. 41, 1263–1285 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  36. He, Y.N., Wang, A.W.: A simplified two-level method for the steady Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 197, 1568–1576 (2008)

  37. Li, Y., Mei, L.Q., Li, Y.Q., Zhao, K.: A two-level variational multiscale method for incompressible flows based on two local Gauss integrations. Numer. Methods Partial Differ. Equ. 29, 1986–2003 (2013)

  38. Shang, Y. Q., Qin, J.: A finite element variational multiscale method based on two-grid discretization for the steady incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 300, 182–198 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  39. Zheng, B., Shang, Y. Q.: A two-level stabilized quadratic equal-order finite element variational multiscale method for incompressible flows. Appl. Math. Comput. 384, 125373 (2020)

    MATH  MathSciNet  Google Scholar 

  40. Layton, W., Lenferink, H.W.J.: A two-level method with backtraking for the Navier-Stokes equations. SIAM J. Numer. Anal. 35, 2035–2054 (1998)

  41. Du, G. Z., Li, Q. T., Zhang, Y. H.: A two-grid method with backtracking for the mixed Navier-Stokes/Darcy model. Numer. Methods Partial Differ. Equ. 36, 1601–1610 (2020)

    Article  MathSciNet  Google Scholar 

  42. Yang, X.C., Shang, Y.Q., Zheng, B.: A simplified two-level subgrid stabilized method with backtracking technique for incompressible flows at high Reynolds numbers. Numer. Methods Partial Differ. Equ. 37, 2067–2088 (2021)

  43. Dai, X. X., Cheng, X. L.: A two-grid method based on Newton iteration for the Navier-Stokes equations. J. Comput. Appl. Math. 220, 566–573 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  44. Huang, P. Z., Feng, X. L.: Two-level stabilized method based on Newton iteration for the steady Smagorinsky model. Nonlinear Anal. Real World Appl. 14, 1795–1805 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  45. Saumya, B., Amiya, K. P.: On three steps two-grid finite element methods for the 2D-transient Navier-Stokes equations. J. Numer. Math. 25, 199–228 (2017)

    MATH  MathSciNet  Google Scholar 

  46. Wang, L., Li, J., Huang, P. Z.: An efficient two-level algorithm for the 2D/3D stationary incompressible magnetohydrodynamics based on the finite element method. Int. Commun. Heat Mass Trans. 98, 183–190 (2018)

    Article  Google Scholar 

  47. Shang, Y. Q.: A new two-level defect-correction method for the steady Navier-Stokes equations. J. Comput. Appl. Math. 381, 113009 (2020)

    Article  MATH  MathSciNet  Google Scholar 

  48. Zheng, H. B., Hou, Y. R., Shi, F., Song, L. N.: A finite element variational multiscale method for incompressible flows based on two local Gauss integrations. J. Comput. Phys. 228, 5961–5977 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  49. Adams, R.: Sobolev Spaces. Academaic Press Inc, New York (1975)

  50. Li, Y., Li, K. T.: Uzawa iteration method for Stokes type variational inequality of the second kind. Acta Math. Appl. Sin. Engl. Ser. 27, 302–315 (2011)

    Article  MathSciNet  Google Scholar 

  51. Girault, V., Raviart, P. A.: Finite Element Method for Navier-Stokes Equations: Theory and Algorithms. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  52. Hood, P., Taylor, C.: A numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids 1, 73–100 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  53. Hecht, F.: New development in FreeFem++. J. Numer. Math. 20, 251–265 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  54. Ghia, U., Ghia, K. N., Shin, C. T.: High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys. 48, 387–441 (1982)

    Article  MATH  Google Scholar 

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Acknowledgements

The authors would like to express their deep gratitude to the anonymous referees for their valuable comments and suggestions, which led to a large improvement of the manuscript.

Funding

This work was supported by the Natural Science Foundation of China (No. 11361016), the Natural Science Foundation of Chongqing Municipality, China (No. cstc2021jcyj-msxmX1044), and Graduate Scientific Research Innovation Project of Chongqing Municipality, China (No. CYB21095).

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  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Bo Zheng & Yueqiang Shang

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  1. Bo Zheng
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Correspondence to Yueqiang Shang.

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Zheng, B., Shang, Y. A three-step defect-correction algorithm for incompressible flows with friction boundary conditions. Numer Algor 91, 1483–1510 (2022). https://doi.org/10.1007/s11075-022-01311-0

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